scholarly journals Nonlinear Periodic Oscillation of a Cylindrical Microvoid Centered at an Isotropic Incompressible Ogden Cylinder

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Wenzheng Zhang ◽  
Xuegang Yuan ◽  
Hongwu Zhang
Keyword(s):  
Blow Up ◽  
Tensile Load ◽  
Periodic Oscillation ◽  
Homoclinic Orbits ◽  
Nonlinear Ordinary Differential Equation ◽  
Incompressibility Condition ◽  
Material Parameters ◽  
Nonlinear Periodic Oscillation ◽  
Increasing Load ◽  
Long Cylinder

We study the dynamic mathematical model for an infinitely long cylinder composed of an isotropic incompressible Ogden material with a microvoid at its center, where the outer surface of the cylinder is subjected to a uniform radial tensile load. Using the incompressibility condition and the boundary conditions, we obtain a second-order nonlinear ordinary differential equation that describes the motion of the microvoid with time. Qualitatively, we find that this equation has two types of solutions. One is a classical nonlinear periodic solution which describes that the motion of the microvoid is a nonlinear periodic oscillation; the other is a blow-up solution. Significantly, for the isotropic incompressible Ogden material, there exist some special values of material parameters, the phase diagrams of the motion equation have homoclinic orbits, which means that the amplitude of a nonlinear periodic oscillation increases discontinuously with the increasing load.

Nonlinear Periodic Oscillations of Micro-Voids in a Class of Incompressible Hyper-Elastic Spheres

2011 ◽  
Vol 52-54 ◽  
pp. 220-225
Author(s):  
Miao Miao Cai ◽  
Jia Na Meng ◽  
Xue Gang Yuan ◽  
Da Tian Niu
Keyword(s):  
Differential Equation ◽  
Oscillation Amplitude ◽  
Tensile Load ◽  
Periodic Oscillation ◽  
Transversely Isotropic ◽  
Material Model ◽  
Symmetric Motion ◽  
Order Ordinary Differential Equation ◽  
Periodic Oscillations ◽  
Hyper Elastic

The problem of radially symmetric motion is examined for a pre-existing micro-void in the interior of a sphere under a suddenly applied outer surface tensile load, where the sphere is composed of a homogeneous incompressible hyper-elastic material. Through qualitatively analyzing the second-order ordinary differential equation that describes the motion of the pre-existing micro-void with time, some interesting conclusions are proposed. For any given values of surface tensile loads, it is proved that the motion of the pre-existing micro-void with time presents a nonlinear periodic oscillation, however, in certain cases, the oscillation amplitude increases discontinuously with the increasing values of surface tensile loads. Finally, based on the known transversely isotropic incompressible Gent-Thomas material model as an example, numerical simulations are carried out.

A Computational Study of Interfacial Stress Distribution in Unidirectional Composites and Its Use in Prediction of Brittle Failure

Aerospace ◽  
2003 ◽  
Author(s):  
Xiaoming Chen ◽  
Thanasis D. Papathanasiou
Keyword(s):  
Brittle Failure ◽  
Composite Structures ◽  
Computational Study ◽  
Tensile Load ◽  
Interfacial Stress ◽  
Uniaxial Tensile ◽  
Unidirectional Composites ◽  
Material Parameters ◽  
Specific Shape ◽  
Random Composite

A computational investigation has been conducted to examine the effect of key microstructural and material parameters, namely the minimum inter-fiber spacing (δ) and the fiber/matrix stiffness ratio (Ef/Em), on the interfacial stress distributions in unidirectional composites subjected to transverse uniaxial tensile load. Representative Volume Elements (RVE’s) containing 144 fibers are constructed using a Monte-Carlo (MC) algorithm, imitating random composite structures. The boundary element method is then used to solve the multi-region elasticity problem on these microstructures. We pay particular attention to the statistics of the distribution of the maximum interfacial stresses computed on each fiber; these are found to follow a Weibull distribution, whose specific shape depends on both microstructural and material parameters. Following the weakest-link theory of Batdorf and Crose [1], we derive a statistical formula for the prediction of brittle failure of composite structures caused by interfacial failure. The limitations and possible extensions to the proposed approach are discussed as well.

Axial–Torsional interactions and wire deformation in 19-wire spiral strand

1988 ◽  
Vol 23 (2) ◽  
pp. 79-86 ◽  
Author(s):  
W S Utting ◽  
N Jones
Keyword(s):  
Tensile Load ◽  
Bending Moment ◽  
Tensile Tests ◽  
Gauge Length ◽  
Strain Gauges ◽  
Bending Moments ◽  
Test Results ◽  
Predicted Values ◽  
Fixed End ◽  
Increasing Load

Tensile tests were performed on a straight steel strand of three layer (12/6/1) construction, having a core wire diameter of 3.66 mm and 3.33 mm diameter helical wires, under conditions of full end-fixity, partial restraint, and ends free from torsional restraint. The torque generated under tensile load was recorded as well as the strand extension and rotation over a 600 mm gauge length. Wire tensions and bending moments in the outer layer of helical wires were determined at the mid-strand position from the outputs of strain gauges in groups of three with parallel grids and mounted parallel to the wire axis on the crown of each wire. The rate of strand extension under tensile load was found to be greater in tests with reduced torsional restraint, the greatest rate occurring in the free-end test. The strand rotation rate was also found to be greatest in the free-end test. The greatest difference from the theoretically predicted rates occurred in a free-end test with increasing load; predicted values of extension and rotation underestimated the test results by 12 and 23 per cent, respectively. Displacement of the load-torque plots occurred in the direction of reducing torque as testing proceeded. This appears to indicate the redistribution of the strand load between the layers of wires. Wire tensions showed a more even sharing of load in the fixed-end condition than in the free-end condition. The increase in rate of tension with strand load was less for most wires in tests with reduced torsional restraint, with the lowest tension rates developing in the free-end condition. For most wires, the rate of bending moment change with strand load was greater (in the sense tending to decrease tensile stress on wire crowns) in tests with reduced torsional restraint. However, the bending moment rates varied greatly between wires, the variation being greater in tests with reduced torsional restraint than in fixed-end tests.

Approximations of the Homoclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two

2017 ◽  
Vol 27 (07) ◽  
pp. 1750109 ◽  
Author(s):  
Carmen Rocşoreanu ◽  
Mihaela Sterpu
Keyword(s):  
Normal Form ◽  
Blow Up ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Second Order ◽  
Dimensional System ◽  
Double Zero ◽  
Codimension Two ◽  
Two Dimensional System ◽  
Codimension Two Bifurcation

The two-dimensional system of differential equations corresponding to the normal form of the double-zero bifurcation with symmetry of order two is considered. This is a codimension two bifurcation. The associated dynamical system exhibits, among others, a homoclinic bifurcation. In this paper, we obtain second order approximations both for the curve of parametric values of homoclinic bifurcation and for the homoclinic orbits. To perform this task, we reduce first the normal form to a perturbed Hamiltonian system, using a blow-up technique. Then, by means of a perturbation method, we determine explicit first and second order approximations of the homoclinic orbits. The solutions obtained theoretically are compared with those obtained numerically for several cases. Finally, an application of the obtained results is presented.

scholarly journals Investigation of Strain of Steel Reinforcement of Modular Flexural Member at Discontinuity Interface

2019 ◽  
Vol 9 (22) ◽  
pp. 4922
Author(s):  
Park ◽  
Lee ◽  
Park ◽  
Choi ◽  
Hong
Keyword(s):  
Crack Width ◽  
Tensile Load ◽  
Tensile Behavior ◽  
Steel Reinforcement ◽  
Modular Structure ◽  
Structural Behavior ◽  
Flexural Member ◽  
Maximum Crack ◽  
Increasing Load

The modular structure has a discontinuity owing to the joint between the modules; thus, structural behavior verification is required. In this study, the tensile behavior of a steel reinforcement at the discontinuity interface was evaluated in the joint of a modular flexural member. The modular specimen was fabricated with a 400 mm joint, and an integral specimen was fabricated with the same specifications as the modular specimen, without a joint. The largest crack width of the integral specimen was measured at the center of the beam, and that of the modular specimen was measured at the discontinuity interface. The maximum crack width of the modular specimen was greater than that of the integral specimen. The strain of the steel reinforcement was estimated using the measured crack width and six formulas for evaluating the crack width. The estimated strain of the modular specimen was higher than that of the integral specimen, and the deformation of the steel reinforcement at the discontinuity interface was accelerated with the increasing load. Therefore, the tensile load was concentrated at the discontinuity interface in the modular specimen, and the steel reinforcement at the discontinuity interface was likely to yield earlier than the integral specimen.

scholarly journals Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model

2019 ◽  
Vol 8 (1) ◽  
pp. 216-223 ◽  
Author(s):  
Kourosh Parand ◽  
Fatemeh Mirahmadian ◽  
Mehdi Delkhosh
Keyword(s):  
Stretching Sheet ◽  
Fluid Model ◽  
Nonlinear Ordinary Differential Equation ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Ordinary Differential Equations ◽  
Material Parameters ◽  
Linear Algebraic Equations ◽  
Legendre Collocation Method ◽  
Layer Flow

Abstract In this paper, a spectral method based on the rational Legendre functions is discussed to approximate the solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a stretching sheet. At first, because of the model is a nonlinear ordinary differential equation, we construct a sequence of linear ordinary differential equations (ODEs) by using the quasilinearization method (QLM). By applying QLM on the ODEs at each iteration, the equations convert to a system of linear algebraic equations. The present results have shown the high attention and convergence of our method, and the residual error of the present results are very small. In addition, the effect of the Eyring-Powell fluid material parameters is explained.

FEM Analysis and Verification of Stress Intensity Factor of Dugdale Model in Silicon Steel Plate under Biaxial Tensile Load

2011 ◽  
Vol 675-677 ◽  
pp. 979-982
Author(s):  
J.G. Wang ◽  
S.L. Li ◽  
Dong Ying Ju ◽  
M.J. Sun
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Tensile Load ◽  
Load Ratio ◽  
Fem Analysis ◽  
Dugdale Model ◽  
Displacement Extrapolation Method ◽  
Increasing Load ◽  
Good Agreement

A plate containing a central crack and subjected to biaxial stresses has been studied by a finite element method using ANSYS. The stress intensity factor was calculated using displacement extrapolation method and Dugdal model. From the results, it was found that the stress intensity factors increased with the increasing load ratio. There was good agreement between experimental and theoretical results.

Influence of grain direction on the time-dependent behavior of wood analyzed by a 3D rheological model. A mathematical consideration

Holzforschung ◽  
2018 ◽  
Vol 72 (10) ◽  
pp. 889-897 ◽  
Author(s):  
Sabina Huč ◽  
Staffan Svensson
Keyword(s):  
Rheological Model ◽  
Tensile Load ◽  
Longitudinal Direction ◽  
Material Behavior ◽  
Time Dependent ◽  
Sustained Load ◽  
Uniaxial Tensile ◽  
Material Parameters ◽  
Dependent Behavior ◽  
Natural Time

AbstractA three-dimensional (3D) rheological model for an orthotropic material subjected to sustained load or deformation under constant climate has been mathematically formulated. The elastic and viscoelastic compliance matrices are symmetric, where the mathematical derivation of the latter is shown. The model is linear and requires constant numerical values for the elastic and viscoelastic material parameters. The model’s ability to predict the natural time-dependent response in three material directions simultaneously is demonstrated on a Douglas fir (Pseudotsuga menziesii) specimen subjected to a constant uniaxial tensile load. The material extends in a longitudinal direction and contracts in the transverse directions with time. The required material parameters are taken from the literature when possible, otherwise they are assumed. Furthermore, the influence of misalignment between the directions of observation and wood material directions on induced time-dependent strains is analyzed. The analyses show that the misalignment has a large effect on the material behavior. In some cases, the specimen under constant uniaxial tension even extends in the perpendicular transverse direction with time. The obtained results clearly demonstrate the high importance of considering the alignment of material directions precisely in order to be able to interpret the time-dependent behavior of wood correctly.

Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem

2017 ◽  
Vol 7 (4) ◽  
pp. 679-696
Author(s):  
Chien-Hong Cho ◽  
Chun-Yi Liu
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Time ◽  
Finite Difference Scheme ◽  
Blow Up ◽  
Nonlinear Ordinary Differential Equation ◽  
Grid Size ◽  
Second Order ◽  
Finite Difference Schemes

AbstractWe consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, whose solution becomes unbounded in a finite time T. The finite time T is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.

Bifurcation Complexity from Orbit-Flip Homoclinic Orbit of Weak Type

2016 ◽  
Vol 26 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Qiuying Lu ◽  
Vincent Naudot
Keyword(s):  
Homoclinic Orbit ◽  
Weak Type ◽  
Blow Up ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Unperturbed System ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Three Dimensional Vector Field ◽  
Degenerate Version

In this paper, we study the unfolding of a three-dimensional vector field having an orbit-flip homoclinic orbit of weak type. Such a homoclinic orbit is a degenerate version of the so-called orbit-flip homoclinic orbit. We show the existence of inclination-flip homoclinic orbits of arbitrary higher order bifurcating from the unperturbed system. Our strategy consists of using the local moving coordinates method and blow up in the parameter space. In addition, the numerical existence of the orbit-flip homoclinic orbit of weak type is presented based on the truncated Taylor expansion and the numerical computation for both the strong stable manifold and unstable manifold.

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